Using the first rule of etymology, that the most colorful origin of a word or phrase becomes accepted, the idea that building a fire to hasten the exit of child chimney sweeps out of chimneys has a certain Dickensian desperation . . . but just a little over the top. The many problems with this hyperbolic notion seem so obvious that it seems unnecessary to point them out, but I will if anyone replies; and I hesitate to change this without other opinions.
Many idioms have multiple equally valid etymologies, depending on when the phrase was in currency or particularly popular, but the one I am most familiar with is building a fire under a mule to get it to move. Mules have a well deserved reputation for stubbornness, and were absolutely necessary for small farms (Cf "40 acres and a mule"), but mules don't want to move, they are able to splay their legs out in such a way that they become functionally immovable. Farmers would literally build a fire under their belly or chests to get them to move, and once incentivized to move, could then be controlled again. It did not take much of a fire, and mules moved as soon as they felt the pain of the fire.
I will check on more printed sources, but there are a number of idioms for which there is no agreement. 3 Richard C. Green (talk) 07:15, 26 April 2023 (UTC)